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[MUSIC PLAYING]
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PROFESSOR: In the last several
lectures, we've talked about a
10
00:00:59,510 --> 00:01:02,930
generalization of the
continuous-time Fourier
11
00:01:02,930 --> 00:01:07,490
transform and a very similar
strategy also applies to
12
00:01:07,490 --> 00:01:10,610
discrete-time, and that's what
we want to begin to deal with
13
00:01:10,610 --> 00:01:12,750
in today's lecture.
14
00:01:12,750 --> 00:01:17,400
So what we want to talk about
is generalizing the Fourier
15
00:01:17,400 --> 00:01:21,650
transform, and what this will
lead to in discrete-time is a
16
00:01:21,650 --> 00:01:24,380
notion referred to as
the z-transform.
17
00:01:24,380 --> 00:01:27,810
Now, just as in continuous-time,
in
18
00:01:27,810 --> 00:01:32,670
discrete-time the Fourier
transform corresponded to a
19
00:01:32,670 --> 00:01:38,830
representation of a sequence
as a linear combination of
20
00:01:38,830 --> 00:01:41,750
complex exponentials.
21
00:01:41,750 --> 00:01:45,430
So this was the synthesis
equation.
22
00:01:45,430 --> 00:01:46,810
And, of course, there is the
23
00:01:46,810 --> 00:01:50,100
corresponding analysis equation.
24
00:01:50,100 --> 00:01:55,310
And as you recall, and as is the
same for continuous-time,
25
00:01:55,310 --> 00:01:59,300
the reason that we picked
complex exponentials was
26
00:01:59,300 --> 00:02:02,900
because of the fact that they
are eigenfunctions of linear
27
00:02:02,900 --> 00:02:04,730
time-invariant systems.
28
00:02:04,730 --> 00:02:09,229
In other words, if you have a
complex exponential into a
29
00:02:09,229 --> 00:02:12,560
linear time-invariant system,
the output is a complex
30
00:02:12,560 --> 00:02:13,820
exponential.
31
00:02:13,820 --> 00:02:19,620
And the change in complex
amplitude, which corresponds
32
00:02:19,620 --> 00:02:23,150
to the frequency response, in
fact is what led to the
33
00:02:23,150 --> 00:02:25,640
definition of the Fourier
transform.
34
00:02:25,640 --> 00:02:29,070
In particular, it is the Fourier
transform of the
35
00:02:29,070 --> 00:02:31,610
impulse response.
36
00:02:31,610 --> 00:02:36,450
Well, that set of notions is,
more or less, identical to the
37
00:02:36,450 --> 00:02:40,410
way we motivated the Laplace
transform in the
38
00:02:40,410 --> 00:02:43,030
continuous-time case, in the
Fourier transform in the
39
00:02:43,030 --> 00:02:44,680
continuous-time case.
40
00:02:44,680 --> 00:02:49,710
And just as in continuous-time,
there are a
41
00:02:49,710 --> 00:02:55,050
set of signals more general than
the complex exponentials,
42
00:02:55,050 --> 00:02:57,760
which are also eigenfunctions
of linear
43
00:02:57,760 --> 00:02:59,420
time-invariant systems.
44
00:02:59,420 --> 00:03:03,810
In particular, in discrete-time,
if we had
45
00:03:03,810 --> 00:03:08,790
instead of an exponential e to
the j omega n, we had a more
46
00:03:08,790 --> 00:03:16,220
general complex number z, that
the signal z to the n is also
47
00:03:16,220 --> 00:03:21,090
an eigenfunction of a linear
time-invariant system for any
48
00:03:21,090 --> 00:03:22,630
particular z.
49
00:03:22,630 --> 00:03:26,770
So we can see that by
substituting that into the
50
00:03:26,770 --> 00:03:30,580
convolution sum and recognizing,
again, very
51
00:03:30,580 --> 00:03:34,820
strongly paralleling the
continuous-time argument, that
52
00:03:34,820 --> 00:03:42,670
we can rewrite this factor as
z to the n z to the minus k.
53
00:03:42,670 --> 00:03:48,960
And because of the fact that
it's a sum on k and this term
54
00:03:48,960 --> 00:03:52,460
doesn't depend on k, we can
take that term out.
55
00:03:52,460 --> 00:03:57,010
And the conclusion is that if
we have z to the n as an
56
00:03:57,010 --> 00:04:04,640
input, that the output is of the
same form times a factor
57
00:04:04,640 --> 00:04:06,980
which depends on z.
58
00:04:06,980 --> 00:04:10,630
But of course, doesn't depend
on k because that is summed
59
00:04:10,630 --> 00:04:13,260
out as we form the summation.
60
00:04:13,260 --> 00:04:18,149
So, in fact, this summation
corresponds to a complex
61
00:04:18,149 --> 00:04:26,490
number, which we'll denote as H
of z, where z will represent
62
00:04:26,490 --> 00:04:28,750
a more general complex number.
63
00:04:28,750 --> 00:04:34,870
Namely, z is r e to the j omega,
r being the magnitude
64
00:04:34,870 --> 00:04:38,020
of this complex number
and omega, of
65
00:04:38,020 --> 00:04:40,530
course, being the angle.
66
00:04:40,530 --> 00:04:46,950
So for a linear time-invariant
system, a more general complex
67
00:04:46,950 --> 00:04:51,900
exponential sequence of this
form generates as an output a
68
00:04:51,900 --> 00:04:55,990
complex exponential sequence of
the same form with a change
69
00:04:55,990 --> 00:05:00,480
in amplitude which we're
representing as H of z,
70
00:05:00,480 --> 00:05:04,010
recognizing the fact that it's
going to be a function of what
71
00:05:04,010 --> 00:05:06,080
that complex number is.
72
00:05:06,080 --> 00:05:11,150
And this amplitude factor is
given by this summation.
73
00:05:11,150 --> 00:05:18,130
And it is this summation which
is defined as the z-transform
74
00:05:18,130 --> 00:05:22,100
of the sequence H of n.
75
00:05:22,100 --> 00:05:24,240
Now, let me stress that--
76
00:05:24,240 --> 00:05:27,650
and I'll continue to stress this
as the lecture goes on.
77
00:05:27,650 --> 00:05:33,260
That much of what we've said is
directly parallel to what
78
00:05:33,260 --> 00:05:35,790
we said in the continuous-time
case.
79
00:05:35,790 --> 00:05:41,310
And what we've simply done
is to expand from complex
80
00:05:41,310 --> 00:05:46,060
exponentials with a purely
imaginary exponent, complex
81
00:05:46,060 --> 00:05:50,950
exponential time functions or
sequences of that form, to
82
00:05:50,950 --> 00:05:56,990
ones that have more general
complex exponential factors.
83
00:05:56,990 --> 00:06:03,830
Now, we have a mapping here from
the impulse response to
84
00:06:03,830 --> 00:06:08,660
the amplitude or the eigenvalue
associated with
85
00:06:08,660 --> 00:06:14,320
that input, and this mapping is
what is referred to as the
86
00:06:14,320 --> 00:06:16,130
z-transform.
87
00:06:16,130 --> 00:06:21,120
So H of z is, in fact,
the z-transform
88
00:06:21,120 --> 00:06:23,060
of the impulse response.
89
00:06:23,060 --> 00:06:27,010
And if we consider applying
this mapping as we did in
90
00:06:27,010 --> 00:06:30,420
continuous-time in a similar
argument, applying this
91
00:06:30,420 --> 00:06:35,490
mapping to a sequence whether
or not it corresponds to the
92
00:06:35,490 --> 00:06:39,180
impulse response or a linear
time-invariant system, that
93
00:06:39,180 --> 00:06:46,270
leads then to the z-transform of
a general sequence x of n.
94
00:06:46,270 --> 00:06:51,410
The z-transform being defined
by this relationship.
95
00:06:51,410 --> 00:06:56,460
And again, notationally, we'll
often represent a time
96
00:06:56,460 --> 00:07:00,740
function and a z-transform
through a shorthand notation,
97
00:07:00,740 --> 00:07:06,690
just indicating that x of z is
the z-transform of x of n.
98
00:07:09,260 --> 00:07:12,750
So we've kind of motivated the
development in a manner
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00:07:12,750 --> 00:07:14,760
exactly identical to
what we had done
100
00:07:14,760 --> 00:07:16,570
with the Laplace transform.
101
00:07:16,570 --> 00:07:19,010
Kind of the idea that if you
look at the eigenvalue
102
00:07:19,010 --> 00:07:23,620
associated with a linear
time-invariant system, that
103
00:07:23,620 --> 00:07:29,220
essentially generates a mapping
between the sequence--
104
00:07:29,220 --> 00:07:33,750
system impulse response
and a function of z.
105
00:07:33,750 --> 00:07:36,690
And that corresponds to the
z-transform here, it
106
00:07:36,690 --> 00:07:39,020
corresponded to the Laplace
transform in the
107
00:07:39,020 --> 00:07:41,760
continuous-time case.
108
00:07:41,760 --> 00:07:46,850
That same argument is also the
kind of argument that we use
109
00:07:46,850 --> 00:07:51,200
to lead us into the Fourier
transform originally.
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00:07:51,200 --> 00:07:57,400
And once again, what you would
expect is that the z-transform
111
00:07:57,400 --> 00:08:00,170
has a very close and important
relationship
112
00:08:00,170 --> 00:08:01,810
to the Fourier transform.
113
00:08:01,810 --> 00:08:06,730
And indeed, that relationship
turns out to be, more or less,
114
00:08:06,730 --> 00:08:11,630
identical to the relationship
between the Laplace transform
115
00:08:11,630 --> 00:08:15,400
and the Fourier transform
in continuous-time.
116
00:08:15,400 --> 00:08:18,650
Well, let's look at
the relationship.
117
00:08:18,650 --> 00:08:23,240
First of all, what we recognize
is that if we
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00:08:23,240 --> 00:08:30,960
compare the Fourier transform
expression for a sequence and
119
00:08:30,960 --> 00:08:36,299
the z-transform expression for
the same sequence that they
120
00:08:36,299 --> 00:08:39,750
involve, essentially,
the same operations.
121
00:08:39,750 --> 00:08:47,230
And in fact, since z is of the
form r e to the j omega, if we
122
00:08:47,230 --> 00:08:52,300
want this sum to look like this
sum, then that would mean
123
00:08:52,300 --> 00:08:56,900
that we would choose z equal
to e to the j omega.
124
00:08:56,900 --> 00:09:01,740
Said another way, the
z-transform, when z is e to
125
00:09:01,740 --> 00:09:04,990
the j omega, is going to reduce
126
00:09:04,990 --> 00:09:07,900
to the Fourier transform.
127
00:09:07,900 --> 00:09:12,290
So we have a relationship like
the one, again, that we had
128
00:09:12,290 --> 00:09:15,140
between the Laplace transform
and the Fourier transform in
129
00:09:15,140 --> 00:09:16,710
continuous-time.
130
00:09:16,710 --> 00:09:22,020
Namely that for a certain set
of values of the complex
131
00:09:22,020 --> 00:09:26,220
variable, the transform, the
z-transform, reduces to the
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00:09:26,220 --> 00:09:28,470
Fourier transform.
133
00:09:28,470 --> 00:09:32,970
So if we have x of z, the
z-transform, and we look at
134
00:09:32,970 --> 00:09:40,100
that for z equal to e to the j
omega, and z equal to e to the
135
00:09:40,100 --> 00:09:43,850
j omega is similar to saying
that we're looking at that for
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00:09:43,850 --> 00:09:46,840
the magnitude of z equal to 1.
137
00:09:46,840 --> 00:09:49,740
We're specifically choosing
r equal to 1, which is the
138
00:09:49,740 --> 00:09:52,050
magnitude of z.
139
00:09:52,050 --> 00:09:54,550
Then this is equal
to the Fourier
140
00:09:54,550 --> 00:09:57,290
transform of the sequence.
141
00:09:57,290 --> 00:10:02,580
So the z-transform for z equal
to e to the j omega is the
142
00:10:02,580 --> 00:10:07,020
Fourier transform,
and so this then
143
00:10:07,020 --> 00:10:11,290
corresponds to x of omega.
144
00:10:11,290 --> 00:10:12,540
Namely, the Fourier transform.
145
00:10:15,000 --> 00:10:19,510
Well, we now have ourselves in a
similar situation, again, to
146
00:10:19,510 --> 00:10:22,840
what we had when we talked about
the Laplace transform.
147
00:10:22,840 --> 00:10:28,050
Namely, a notational
awkwardness, or inconvenience,
148
00:10:28,050 --> 00:10:30,510
which we can resolve by simply
149
00:10:30,510 --> 00:10:33,840
redefining some of our notation.
150
00:10:33,840 --> 00:10:36,960
In particular, the awkwardness
relates to the fact that
151
00:10:36,960 --> 00:10:40,990
whereas we've been writing our
Fourier transforms this way,
152
00:10:40,990 --> 00:10:46,620
as x of omega, if we were to
express x of z and look at it,
153
00:10:46,620 --> 00:10:48,510
it's equal to e to
the j omega.
154
00:10:48,510 --> 00:10:52,320
We end up with the independent
variable being e to the j
155
00:10:52,320 --> 00:10:56,240
omega rather than omega.
156
00:10:56,240 --> 00:10:58,920
Well, in fact, the Fourier
transform is
157
00:10:58,920 --> 00:11:00,020
a function of omega.
158
00:11:00,020 --> 00:11:02,620
It's also a function of
e to the j omega.
159
00:11:02,620 --> 00:11:06,620
And now what we can see is that
given the fact that we
160
00:11:06,620 --> 00:11:12,310
want to generalize the Fourier
transform to the z-transform,
161
00:11:12,310 --> 00:11:17,870
it's convenient now to use as
notation for the Fourier
162
00:11:17,870 --> 00:11:22,120
transform x of z with z equal
to e to the j omega.
163
00:11:22,120 --> 00:11:25,030
Namely, our Fourier transforms
will now be written as I've
164
00:11:25,030 --> 00:11:26,640
indicated here.
165
00:11:26,640 --> 00:11:31,970
So just summarizing that, our
new notation is that the
166
00:11:31,970 --> 00:11:37,980
independent variable on the
Fourier transform is now going
167
00:11:37,980 --> 00:11:42,480
to be expressed as e to the j
omega rather than as omega.
168
00:11:42,480 --> 00:11:47,320
It's a minor notational change,
but I recognize the
169
00:11:47,320 --> 00:11:51,630
fact that it's somewhat
confusing initially, and takes
170
00:11:51,630 --> 00:11:55,200
a few minutes to sit down and
just get it straightened out.
171
00:11:55,200 --> 00:11:56,580
It's very similar
to what we did
172
00:11:56,580 --> 00:11:58,440
with the Laplace transform.
173
00:11:58,440 --> 00:12:02,560
But let me draw your attention
to the fact that in the
174
00:12:02,560 --> 00:12:06,090
Laplace transform, the
independent variable that we
175
00:12:06,090 --> 00:12:11,520
ended up with in talking about
the Fourier transform is
176
00:12:11,520 --> 00:12:14,640
different than what we're
ending up with here.
177
00:12:14,640 --> 00:12:18,160
In particular, before we
had j omega, now we
178
00:12:18,160 --> 00:12:21,430
have e to the j omega.
179
00:12:21,430 --> 00:12:27,350
And the reason for that is
simply that whereas in
180
00:12:27,350 --> 00:12:29,530
continuous time we were talking
about functions of the
181
00:12:29,530 --> 00:12:34,530
form e to the st, now we're
talking about sequences of the
182
00:12:34,530 --> 00:12:38,100
form z to the n.
183
00:12:38,100 --> 00:12:41,940
So we have one relationship
between the Fourier transform
184
00:12:41,940 --> 00:12:43,060
and the z-transform.
185
00:12:43,060 --> 00:12:47,330
Namely, the fact that for the
magnitude of z equal to 1, the
186
00:12:47,330 --> 00:12:53,390
z-transform reduces to the
Fourier transform.
187
00:12:53,390 --> 00:12:56,690
Now, in the Laplace transform,
we also had another important
188
00:12:56,690 --> 00:13:03,250
relationship and observation,
which was the fact that the
189
00:13:03,250 --> 00:13:07,100
Laplace transform
was the Fourier
190
00:13:07,100 --> 00:13:11,870
transform of x of t modified.
191
00:13:11,870 --> 00:13:12,970
And how was it modified?
192
00:13:12,970 --> 00:13:18,270
It was modified by multiplying
by a decaying or growing
193
00:13:18,270 --> 00:13:21,210
exponential, depending on what
the real part of s is.
194
00:13:21,210 --> 00:13:23,820
Well, we have a very similar
situation with the
195
00:13:23,820 --> 00:13:25,430
z-transform.
196
00:13:25,430 --> 00:13:31,560
In particular, in addition to
the fact that the z-transform
197
00:13:31,560 --> 00:13:34,910
for z equal to e to the j omega
reduces to the Fourier
198
00:13:34,910 --> 00:13:39,670
transform, we'll see that the
z-transform for other values
199
00:13:39,670 --> 00:13:45,330
of z is the Fourier transform
of the sequence with an
200
00:13:45,330 --> 00:13:46,800
exponential weighting.
201
00:13:46,800 --> 00:13:50,180
And let's see where
that comes from.
202
00:13:50,180 --> 00:13:53,620
Here we have the general
expression for the
203
00:13:53,620 --> 00:13:55,230
z-transform.
204
00:13:55,230 --> 00:13:59,680
And recognizing that z is a
complex number which we're
205
00:13:59,680 --> 00:14:05,440
expressing in polar form as r e
to the j omega, substituting
206
00:14:05,440 --> 00:14:10,400
that in, this summation now
becomes x of n, r e to the j
207
00:14:10,400 --> 00:14:12,240
omega to the minus n.
208
00:14:12,240 --> 00:14:17,355
We can factor out these two
terms, r to the minus n and e
209
00:14:17,355 --> 00:14:19,670
to the minus j omega n.
210
00:14:19,670 --> 00:14:26,060
And combining the r to the minus
n with x of n and the e
211
00:14:26,060 --> 00:14:30,130
to the minus j omega n being
treated separately, what we
212
00:14:30,130 --> 00:14:35,530
end up with is the summation
that I have here.
213
00:14:35,530 --> 00:14:39,840
Well, what this says is that
the z-transform, which is
214
00:14:39,840 --> 00:14:46,430
this, at z equal to r e to the
j omega, is in fact the
215
00:14:46,430 --> 00:14:53,200
Fourier transform of what?
216
00:14:53,200 --> 00:14:58,670
It's the Fourier transform
of x of n multiplied by r
217
00:14:58,670 --> 00:15:00,610
to the minus n.
218
00:15:00,610 --> 00:15:06,200
So that is the expression
that we have here.
219
00:15:06,200 --> 00:15:10,150
And in continuous-time, we had
the Laplace transform as the
220
00:15:10,150 --> 00:15:15,710
Fourier transform of x of t
e to the minus sigma t.
221
00:15:15,710 --> 00:15:18,820
Here we have the Fourier
transform of x of n r
222
00:15:18,820 --> 00:15:21,790
to the minus n.
223
00:15:21,790 --> 00:15:25,095
Now, something to just reflect
on for a minute is--
224
00:15:27,730 --> 00:15:31,980
because it tends to cause a
little bit of problem with the
225
00:15:31,980 --> 00:15:35,640
algebra later on if you're
attention isn't drawn to it,
226
00:15:35,640 --> 00:15:39,670
is that we're talking about
multiplying x of n times r to
227
00:15:39,670 --> 00:15:41,410
the minus n.
228
00:15:41,410 --> 00:15:50,120
The question is, for r greater
than 1, does r to the minus n
229
00:15:50,120 --> 00:15:55,220
increase exponentially as n
increases or does it decrease?
230
00:15:55,220 --> 00:15:57,670
We're talking about
r to the minus n.
231
00:15:57,670 --> 00:16:00,860
If r is greater than 1, if
the magnitude of r is
232
00:16:00,860 --> 00:16:01,370
greater than 1.
233
00:16:01,370 --> 00:16:04,760
For example, if it's equal
to 2, r to the minus n
234
00:16:04,760 --> 00:16:06,800
is 1/2 to the n.
235
00:16:06,800 --> 00:16:10,440
And so, in fact, that decreases
exponentially.
236
00:16:10,440 --> 00:16:14,900
Or more generally, the larger
r is, the faster r to the
237
00:16:14,900 --> 00:16:17,930
minus n decays with
increasing n.
238
00:16:22,060 --> 00:16:28,490
Well, let's just look at some
examples of the z-transform.
239
00:16:28,490 --> 00:16:33,930
And examples that I've picked,
again, are examples directly
240
00:16:33,930 --> 00:16:34,820
out of the text.
241
00:16:34,820 --> 00:16:40,900
And so the details of the
algebra you can look at more
242
00:16:40,900 --> 00:16:44,280
leisurely as you sit
with the textbook.
243
00:16:44,280 --> 00:16:48,200
Let's consider, first of all, an
exponential sequence x of n
244
00:16:48,200 --> 00:16:50,960
equals a to the n times
the unit step.
245
00:16:50,960 --> 00:16:56,460
So 0 for negative time and an
exponential for positive time.
246
00:16:56,460 --> 00:17:00,060
And the Fourier transform,
as we've seen in earlier
247
00:17:00,060 --> 00:17:05,210
lectures, is 1 over 1 minus
a e to the minus j omega.
248
00:17:05,210 --> 00:17:08,540
But this doesn't always
converge.
249
00:17:08,540 --> 00:17:12,210
In particular, for convergence
of the Fourier transform, we
250
00:17:12,210 --> 00:17:14,810
would require absolute
summability of
251
00:17:14,810 --> 00:17:16,770
the original sequence.
252
00:17:16,770 --> 00:17:20,319
And that, in turn, requires
that the magnitude of a be
253
00:17:20,319 --> 00:17:22,250
less than 1.
254
00:17:22,250 --> 00:17:25,730
So the Fourier transform is
this, provided that the
255
00:17:25,730 --> 00:17:28,140
magnitude of a is less than 1.
256
00:17:28,140 --> 00:17:31,530
And what is the Fourier
transform if the magnitude of
257
00:17:31,530 --> 00:17:34,330
a is not less than 1?
258
00:17:34,330 --> 00:17:36,380
Well, the answer is
that, in that
259
00:17:36,380 --> 00:17:37,630
case, it doesn't converge.
260
00:17:40,310 --> 00:17:42,740
Now, let's look at
the z-transform.
261
00:17:42,740 --> 00:17:47,770
The z-transform is the sum from
minus infinity to plus
262
00:17:47,770 --> 00:17:54,420
infinity of a to the
n z to the minus n
263
00:17:54,420 --> 00:17:56,480
times the unit step.
264
00:17:56,480 --> 00:17:59,130
The unit step will change
the lower limit to 0.
265
00:17:59,130 --> 00:18:01,460
So it's the sum from
0 to infinity.
266
00:18:01,460 --> 00:18:08,720
And this is of the form
a times z to the
267
00:18:08,720 --> 00:18:11,490
minus 1 to the n.
268
00:18:11,490 --> 00:18:14,740
So we're summing from 0 to
infinity a times z to the
269
00:18:14,740 --> 00:18:16,360
minus 1 to the n.
270
00:18:16,360 --> 00:18:22,320
That sum is 1 over 1 minus
a z to the minus 1.
271
00:18:22,320 --> 00:18:29,000
But in order for that sum to
converge, we require that the
272
00:18:29,000 --> 00:18:35,710
magnitude of a times z to the
minus 1 be less than 1.
273
00:18:35,710 --> 00:18:43,320
Now, the z-transform is the
Fourier transform of the
274
00:18:43,320 --> 00:18:48,090
sequence a to the n times
r to the minus n.
275
00:18:48,090 --> 00:18:55,530
And this statement about the
z-transform converging is
276
00:18:55,530 --> 00:18:58,710
exactly identical to the
statement that what we're
277
00:18:58,710 --> 00:19:04,280
requiring is that the magnitude
of a times r to the
278
00:19:04,280 --> 00:19:11,350
minus 1 be less than 1, where
this represents the
279
00:19:11,350 --> 00:19:17,440
exponential factor that we have
that in effect is applied
280
00:19:17,440 --> 00:19:21,800
to the sequence, so that the
Fourier transform becomes the
281
00:19:21,800 --> 00:19:23,430
z-transform.
282
00:19:23,430 --> 00:19:27,560
And so, if we put this
condition, we can interpret
283
00:19:27,560 --> 00:19:32,100
this condition in exactly the
same way that we interpret the
284
00:19:32,100 --> 00:19:36,920
condition on convergence of
the Fourier transform.
285
00:19:36,920 --> 00:19:44,230
So from what we've worked out
here then, what we have is the
286
00:19:44,230 --> 00:19:50,160
z-transform of a to the n times
u of n is 1 over 1 minus
287
00:19:50,160 --> 00:19:53,600
a z to the minus 1.
288
00:19:53,600 --> 00:19:58,940
That works for any value of a
provided that we pick the
289
00:19:58,940 --> 00:20:01,550
value of z correctly.
290
00:20:01,550 --> 00:20:05,120
In particular, we have to
pick the set of values
291
00:20:05,120 --> 00:20:07,900
of z, so that what?
292
00:20:07,900 --> 00:20:11,870
So that the magnitude of a
times z to the minus 1
293
00:20:11,870 --> 00:20:13,590
is less than 1.
294
00:20:13,590 --> 00:20:17,640
Or equivalently, so that the
magnitude of z is greater than
295
00:20:17,640 --> 00:20:20,040
the magnitude of a.
296
00:20:20,040 --> 00:20:25,260
So associated with the
z-transform of this sequence
297
00:20:25,260 --> 00:20:29,890
is this algebraic expression,
and this set of values on z
298
00:20:29,890 --> 00:20:32,820
for which that algebraic
expression is valid.
299
00:20:32,820 --> 00:20:38,600
And just as with the Laplace
transform, this range of
300
00:20:38,600 --> 00:20:44,430
values is referred to as the
region of convergence of the
301
00:20:44,430 --> 00:20:45,680
z-transform.
302
00:20:47,530 --> 00:20:52,950
Now, again, as we saw with the
Laplace transform, it's
303
00:20:52,950 --> 00:20:58,590
important to recognize that in
specifying or having worked
304
00:20:58,590 --> 00:21:02,870
out the z-transform of a
sequence, it's not just the
305
00:21:02,870 --> 00:21:07,160
algebraic expression, but also
the region of convergence
306
00:21:07,160 --> 00:21:11,700
that's required to uniquely
specify it.
307
00:21:11,700 --> 00:21:15,420
To emphasize that further,
here is Example
308
00:21:15,420 --> 00:21:17,500
10.2 from the text.
309
00:21:17,500 --> 00:21:20,910
And if you work that one
through, what you find is
310
00:21:20,910 --> 00:21:26,190
that, algebraically, the
z-transform of this sequence
311
00:21:26,190 --> 00:21:30,770
is 1 over 1 minus a
z to the minus 1.
312
00:21:30,770 --> 00:21:34,240
Identical algebraically to
what we had up here.
313
00:21:34,240 --> 00:21:42,360
But now with a region of
convergence, which is the
314
00:21:42,360 --> 00:21:45,640
magnitude of z less than
the magnitude of a.
315
00:21:45,640 --> 00:21:49,470
In contrast to this example,
where the region of
316
00:21:49,470 --> 00:21:53,040
convergence was the magnitude
of z greater than the
317
00:21:53,040 --> 00:21:56,160
magnitude of a.
318
00:21:56,160 --> 00:22:01,170
So again, it requires not just
the algebraic expression, but
319
00:22:01,170 --> 00:22:06,230
also requires a specification of
the region of convergence.
320
00:22:06,230 --> 00:22:12,380
And also, as with the Laplace
transform, it's convenient in
321
00:22:12,380 --> 00:22:19,870
looking at the z-transform
to represent it
322
00:22:19,870 --> 00:22:22,330
in the complex plane.
323
00:22:22,330 --> 00:22:25,710
In this case, the complex
plane referred to as the
324
00:22:25,710 --> 00:22:28,790
z-plane, whereas in
continuous-time when we talked
325
00:22:28,790 --> 00:22:31,400
about the Laplace transform,
it was the s-plane.
326
00:22:31,400 --> 00:22:34,660
z, of course, because z is the
complex variable in terms of
327
00:22:34,660 --> 00:22:37,910
which we're representing
the z-transform.
328
00:22:37,910 --> 00:22:44,090
So we will be representing the
z-transform in terms of
329
00:22:44,090 --> 00:22:48,400
representations in the complex
plane, real part
330
00:22:48,400 --> 00:22:50,530
and imaginary part.
331
00:22:50,530 --> 00:22:54,630
But I've also identified
a circle here.
332
00:22:54,630 --> 00:22:56,830
And you could wonder,
well, what's the
333
00:22:56,830 --> 00:22:59,750
significance of the circle?
334
00:22:59,750 --> 00:23:05,710
Recall that in the discussion
that we just came from, when
335
00:23:05,710 --> 00:23:09,380
we talked about the relationship
between the
336
00:23:09,380 --> 00:23:13,460
z-transform and the Fourier
transform, the z-transform
337
00:23:13,460 --> 00:23:16,290
reduces to the Fourier transform
when the magnitude
338
00:23:16,290 --> 00:23:19,620
of z is equal to 1.
339
00:23:19,620 --> 00:23:21,420
The magnitude of z
equal to 1 in the
340
00:23:21,420 --> 00:23:23,610
complex plane is a circle.
341
00:23:23,610 --> 00:23:28,880
And that circle, in fact,
is a circle of radius 1.
342
00:23:28,880 --> 00:23:33,370
And so it's on this contour
in the z-plane that the
343
00:23:33,370 --> 00:23:38,820
z-transform reduces to the
Fourier transform.
344
00:23:38,820 --> 00:23:40,760
And we'll see some additional
significance of
345
00:23:40,760 --> 00:23:43,550
that as we go along.
346
00:23:43,550 --> 00:23:47,010
Just again to emphasize the
relationships and differences
347
00:23:47,010 --> 00:23:51,150
with continuous-time, with the
Laplace transform it's the
348
00:23:51,150 --> 00:23:56,430
behavior in the s-plane on the
j omega axis that corresponds
349
00:23:56,430 --> 00:23:58,440
to the Fourier transform.
350
00:23:58,440 --> 00:24:02,440
Here it's the behavior on the
unit circle where the
351
00:24:02,440 --> 00:24:07,570
z-transform corresponds to
the Fourier transform.
352
00:24:07,570 --> 00:24:08,930
Now, we'll be talking--
353
00:24:08,930 --> 00:24:12,460
as we did with the Laplace
transform, we'll be talking
354
00:24:12,460 --> 00:24:18,140
very often about transforms
which are rational, and
355
00:24:18,140 --> 00:24:23,820
rational transforms as we'll
see, represent systems which
356
00:24:23,820 --> 00:24:25,960
are characterized by linear
constant coefficient
357
00:24:25,960 --> 00:24:27,700
difference equations.
358
00:24:27,700 --> 00:24:35,510
And so for the rational
z-transforms, we'll again find
359
00:24:35,510 --> 00:24:38,530
it convenient to use a
representation in terms of
360
00:24:38,530 --> 00:24:40,870
poles and zeroes
in the z-plane.
361
00:24:40,870 --> 00:24:45,420
So let's look at our example
as we've worked it out
362
00:24:45,420 --> 00:24:47,920
previously, Example 10.1.
363
00:24:47,920 --> 00:24:54,170
And with this sequence, the
z-transform is 1 divided by 1
364
00:24:54,170 --> 00:24:56,360
minus a z to the minus 1.
365
00:24:56,360 --> 00:24:59,350
And we happen to have written
it as a function of z
366
00:24:59,350 --> 00:25:00,950
to the minus 1.
367
00:25:00,950 --> 00:25:06,200
Clearly, we can rewrite this by
multiplying numerator and
368
00:25:06,200 --> 00:25:09,900
denominator by z, and this would
equivalently then be z
369
00:25:09,900 --> 00:25:13,470
divided by z minus a.
370
00:25:13,470 --> 00:25:19,220
And so if we were to represent
this through a pole-zero plot,
371
00:25:19,220 --> 00:25:22,370
we would have a 0 at the origin
corresponding to this
372
00:25:22,370 --> 00:25:27,340
factor and a pole at z equals
a corresponding to the
373
00:25:27,340 --> 00:25:29,270
denominator factor.
374
00:25:29,270 --> 00:25:34,730
And so the pole-zero pattern for
this is then a pole at z
375
00:25:34,730 --> 00:25:40,190
equal to a and a 0
at the origin.
376
00:25:40,190 --> 00:25:44,415
Now, let me just comment quickly
about the fact that we
377
00:25:44,415 --> 00:25:48,440
had written this as 1 over 1
minus a z to the minus 1, and
378
00:25:48,440 --> 00:25:51,730
that seems kind of strange
because perhaps we should have
379
00:25:51,730 --> 00:25:54,160
multiplied through by z.
380
00:25:54,160 --> 00:25:58,350
Let me just indicate that as
you'll see as you work
381
00:25:58,350 --> 00:26:03,680
examples, it's very typical for
the z-transform to come
382
00:26:03,680 --> 00:26:06,910
out as a function of
z to the minus 1.
383
00:26:06,910 --> 00:26:11,820
And so very typically, you'll
get to recognize that things
384
00:26:11,820 --> 00:26:15,980
will be expressed in terms of
factors involving terms like 1
385
00:26:15,980 --> 00:26:20,380
minus a z to the minus 1, rather
than factors of the
386
00:26:20,380 --> 00:26:23,320
form z minus a.
387
00:26:23,320 --> 00:26:28,740
Well, here is the one example
that we had referred to.
388
00:26:28,740 --> 00:26:33,910
And if we consider another
example, the other example,
389
00:26:33,910 --> 00:26:40,780
which was example 10.2 consists
of an algebraic
390
00:26:40,780 --> 00:26:43,640
expression as I indicate here.
391
00:26:43,640 --> 00:26:47,660
But its region of validity is
the magnitude of z less than
392
00:26:47,660 --> 00:26:49,240
the magnitude of a.
393
00:26:49,240 --> 00:26:53,850
And that corresponds to the
same pole-zero plot, but a
394
00:26:53,850 --> 00:26:59,230
region of convergence which
is inside this circle.
395
00:26:59,230 --> 00:27:05,400
Whereas, in the previous case,
with the pole-zero plot, we
396
00:27:05,400 --> 00:27:09,010
had a region of convergence
which was for the magnitude of
397
00:27:09,010 --> 00:27:11,740
z greater than the
magnitude of a.
398
00:27:11,740 --> 00:27:15,610
So these two examples, this one
and the other one, have
399
00:27:15,610 --> 00:27:18,350
exactly the same pole-zero
pattern and they're
400
00:27:18,350 --> 00:27:22,530
distinguished by their region
of convergence.
401
00:27:22,530 --> 00:27:29,210
Now, notice incidentally that
in this particular case, the
402
00:27:29,210 --> 00:27:32,810
region of convergence includes
the unit circle provided that
403
00:27:32,810 --> 00:27:35,240
the magnitude of a
is less than 1.
404
00:27:35,240 --> 00:27:41,470
And so, in fact, that would say
that the sequence has a
405
00:27:41,470 --> 00:27:43,460
Fourier transform
that converges.
406
00:27:43,460 --> 00:27:47,410
Namely, with the magnitude
of z equal to 1.
407
00:27:47,410 --> 00:27:52,020
Whereas, in this example, the
region of convergence does not
408
00:27:52,020 --> 00:27:54,140
include the unit circle.
409
00:27:54,140 --> 00:27:57,540
And so, in fact, we cannot
look at x of z for the
410
00:27:57,540 --> 00:27:59,570
magnitude of z equal to 1.
411
00:27:59,570 --> 00:28:03,850
And so this example, with the
magnitude of a less than 1,
412
00:28:03,850 --> 00:28:07,950
does not have a Fourier
transform that converges.
413
00:28:07,950 --> 00:28:12,160
Well, assuming that the
magnitude of a is less than 1
414
00:28:12,160 --> 00:28:16,010
and the Fourier transform
converges, we can, in fact,
415
00:28:16,010 --> 00:28:22,080
look at the Fourier transform
by observing what happens as
416
00:28:22,080 --> 00:28:24,460
we go around the unit circle.
417
00:28:24,460 --> 00:28:26,910
We had seen this with the
Laplace transform in terms of
418
00:28:26,910 --> 00:28:31,010
observing what happened as we
move along the j omega axis.
419
00:28:31,010 --> 00:28:39,620
And here again, we can use the
vectors as we trace out the
420
00:28:39,620 --> 00:28:41,060
unit circle.
421
00:28:41,060 --> 00:28:43,790
And in particular, what we would
be looking at in this
422
00:28:43,790 --> 00:28:51,510
case is the ratio of the zero
vector to the pole vector.
423
00:28:51,510 --> 00:28:53,960
For example, if we were looking
at the magnitude of
424
00:28:53,960 --> 00:28:58,120
the z-transform, the magnitude
of the z-transform would be
425
00:28:58,120 --> 00:29:02,520
the ratio of the length of this
vector to the length of
426
00:29:02,520 --> 00:29:04,200
this vector.
427
00:29:04,200 --> 00:29:08,130
And to observe the Fourier
transform, we would observe
428
00:29:08,130 --> 00:29:13,090
how those vectors change in
length as we move around the
429
00:29:13,090 --> 00:29:15,030
unit circle.
430
00:29:15,030 --> 00:29:18,030
And as we move around the unit
circle, what we would trace
431
00:29:18,030 --> 00:29:21,730
out in terms of the ratio of the
lengths of those vectors
432
00:29:21,730 --> 00:29:23,835
is the Fourier transform.
433
00:29:26,910 --> 00:29:36,690
Well, let's focus on that also
in the context of a slightly
434
00:29:36,690 --> 00:29:37,855
different z-transform.
435
00:29:37,855 --> 00:29:43,260
In the z-transform here as we'll
see in a later lecture,
436
00:29:43,260 --> 00:29:45,800
is the z-transform associated
with a second
437
00:29:45,800 --> 00:29:48,070
order difference equation.
438
00:29:48,070 --> 00:29:52,460
It has a denominator factor
which has two poles
439
00:29:52,460 --> 00:29:54,810
associated with it.
440
00:29:54,810 --> 00:30:00,730
And so here, if we assumed that
the Fourier transform of
441
00:30:00,730 --> 00:30:05,360
the associated sequence
converged, then again we would
442
00:30:05,360 --> 00:30:08,080
look at the behavior
of this as we moved
443
00:30:08,080 --> 00:30:10,040
around the unit circle.
444
00:30:10,040 --> 00:30:14,730
And the ratio of the lengths
of the appropriate vectors
445
00:30:14,730 --> 00:30:19,060
would describe for us the
frequency response.
446
00:30:19,060 --> 00:30:21,220
I'm sorry, the Fourier
transform.
447
00:30:21,220 --> 00:30:27,800
So the Fourier transform
magnitude would consist of the
448
00:30:27,800 --> 00:30:32,830
ratio of the lengths of the zero
vectors divided by the
449
00:30:32,830 --> 00:30:34,930
lengths of the pole vectors.
450
00:30:34,930 --> 00:30:39,350
And one thing that we observe
is that as we move in
451
00:30:39,350 --> 00:30:44,290
frequency omega in the vicinity
of this pole, this
452
00:30:44,290 --> 00:30:49,260
pole vector, in fact, reaches
a minimum length.
453
00:30:49,260 --> 00:30:54,170
That would mean that it's
reciprocal would be maximum.
454
00:30:54,170 --> 00:30:57,290
And then, as we sweep past,
the lengths of these two
455
00:30:57,290 --> 00:30:58,380
vectors would increase.
456
00:30:58,380 --> 00:31:02,040
The zero vectors, of course,
would retain the same length
457
00:31:02,040 --> 00:31:05,940
no matter where we were
on the unit circle.
458
00:31:05,940 --> 00:31:08,500
So, in fact, if we looked
at the Fourier transform
459
00:31:08,500 --> 00:31:12,320
associated with this pole-zero
pattern, if this was, for
460
00:31:12,320 --> 00:31:20,760
example, represented the
z-transform of the impulse
461
00:31:20,760 --> 00:31:24,220
response or a linear
time-invariant system, the
462
00:31:24,220 --> 00:31:27,040
corresponding frequency response
would be what I
463
00:31:27,040 --> 00:31:28,970
plotted out below.
464
00:31:28,970 --> 00:31:31,810
And so it would peak.
465
00:31:31,810 --> 00:31:35,560
And in fact, where it would peak
is in the vicinity of the
466
00:31:35,560 --> 00:31:42,220
frequency location of the pole
as I indicate up here.
467
00:31:45,950 --> 00:31:51,530
So as we sweep past this pole
then, in fact, this Fourier
468
00:31:51,530 --> 00:31:52,560
transform .
469
00:31:52,560 --> 00:31:54,550
Peaks.
470
00:31:54,550 --> 00:31:59,060
Well, this notion of looking at
the frequency response as
471
00:31:59,060 --> 00:32:04,210
we move around the unit circle
is a very important notion.
472
00:32:04,210 --> 00:32:06,610
And it's important to recognize
it's the unit circle
473
00:32:06,610 --> 00:32:10,060
we're talking about here,
whereas before we were talking
474
00:32:10,060 --> 00:32:12,740
about the j omega axis.
475
00:32:12,740 --> 00:32:18,470
And to emphasize this further,
let me just show this example.
476
00:32:18,470 --> 00:32:22,900
And in fact, the previous
example with the computer
477
00:32:22,900 --> 00:32:26,730
displays, so that we can see the
frequency response as it
478
00:32:26,730 --> 00:32:31,100
sweeps out as we go around
the unit circle.
479
00:32:31,100 --> 00:32:36,350
So here we have the pole-zero
pattern for the
480
00:32:36,350 --> 00:32:38,640
second order example.
481
00:32:38,640 --> 00:32:42,780
And to generate the Fourier
transform, we want to look at
482
00:32:42,780 --> 00:32:46,360
the behavior of the pole and
zero vectors as we move around
483
00:32:46,360 --> 00:32:47,970
the unit circle.
484
00:32:47,970 --> 00:32:51,200
So first, let's display
the vectors.
485
00:32:51,200 --> 00:32:55,030
And here we have them displayed
to the point
486
00:32:55,030 --> 00:32:58,250
corresponding to
zero frequency.
487
00:32:58,250 --> 00:33:03,750
And the magnitude of the Fourier
transform will be, as
488
00:33:03,750 --> 00:33:06,650
we discussed, the magnitude of
the length of the zero vector
489
00:33:06,650 --> 00:33:08,380
is divided by the
magnitude of the
490
00:33:08,380 --> 00:33:10,580
length of the pole vectors.
491
00:33:10,580 --> 00:33:14,760
Shown below will be the
Fourier transform.
492
00:33:14,760 --> 00:33:18,440
And we have the Fourier
transform displayed here from
493
00:33:18,440 --> 00:33:23,300
0 to 2 pi, rather than from
minus pi to pi as it was
494
00:33:23,300 --> 00:33:25,640
displayed in the transparency.
495
00:33:25,640 --> 00:33:29,130
Because of the periodicity of
the Fourier transform, both of
496
00:33:29,130 --> 00:33:32,440
those are equivalent.
497
00:33:32,440 --> 00:33:38,400
Now we're sweeping away from
omega equals 0 and the lengths
498
00:33:38,400 --> 00:33:40,590
of the pole vectors
have changed.
499
00:33:40,590 --> 00:33:43,100
And that, of course,
generates a change
500
00:33:43,100 --> 00:33:46,390
in the Fourier transform.
501
00:33:46,390 --> 00:33:51,530
And as we continued the
process further, if we
502
00:33:51,530 --> 00:33:57,440
increase frequency, as we sweep
closer to the location
503
00:33:57,440 --> 00:34:02,060
of the pole, the pole vector
decreases in length
504
00:34:02,060 --> 00:34:03,370
dramatically.
505
00:34:03,370 --> 00:34:07,400
And that generates a residence
in the Fourier transform, very
506
00:34:07,400 --> 00:34:11,040
similar to what we saw
in continuous-time.
507
00:34:11,040 --> 00:34:16,875
Now as we continue to sweep
further, what will happen is
508
00:34:16,875 --> 00:34:19,949
that that pole vector
will begin to
509
00:34:19,949 --> 00:34:22,050
increase in length again.
510
00:34:22,050 --> 00:34:25,300
And so, in fact, the magnitude
of the Fourier
511
00:34:25,300 --> 00:34:28,730
transform will decrease.
512
00:34:28,730 --> 00:34:32,889
And we see that here
as we sweep toward
513
00:34:32,889 --> 00:34:35,250
omega equal to pi.
514
00:34:35,250 --> 00:34:39,830
Now, notice in this process that
the length of the zero
515
00:34:39,830 --> 00:34:44,929
vectors has stayed the same
because of the fact that the
516
00:34:44,929 --> 00:34:49,429
zeroes are at the origin, and no
matter where we are in the
517
00:34:49,429 --> 00:34:52,420
unit circle, the length of
those vectors is unity.
518
00:34:52,420 --> 00:34:55,650
So they don't influence in this
example the magnitude,
519
00:34:55,650 --> 00:34:59,420
but they would, of course,
influence the phase.
520
00:34:59,420 --> 00:35:02,600
Now we want to continue sweeping
from omega equal to
521
00:35:02,600 --> 00:35:04,960
pi around to 2 pi.
522
00:35:04,960 --> 00:35:08,280
And because of the symmetry in
the Fourier transform, what we
523
00:35:08,280 --> 00:35:12,310
will see in the magnitude is
identical to what we would see
524
00:35:12,310 --> 00:35:16,880
if we swept from omega equal
to pi back clockwise to
525
00:35:16,880 --> 00:35:19,630
omega equals 0.
526
00:35:19,630 --> 00:35:23,670
In particular now, as we're
increasing frequency, notice
527
00:35:23,670 --> 00:35:27,550
that the length of the pole
vector associated with the
528
00:35:27,550 --> 00:35:30,740
lower half plane pole
is decreasing.
529
00:35:30,740 --> 00:35:35,370
And so, in fact, that
corresponds to generating a
530
00:35:35,370 --> 00:35:39,540
resonance as we sweep
past that pole
531
00:35:39,540 --> 00:35:42,630
location as we are here.
532
00:35:42,630 --> 00:35:46,380
And then finally, that pole
vector increases in length as
533
00:35:46,380 --> 00:35:49,760
we begin to approach omega
equal to 2 pi.
534
00:35:49,760 --> 00:35:53,700
Or equivalently, as we approach
omega equal to 0.
535
00:35:59,400 --> 00:36:05,610
Now finally, let's also look
at the Fourier transform
536
00:36:05,610 --> 00:36:08,670
associated with the first
order example that we
537
00:36:08,670 --> 00:36:11,370
discussed earlier
in the lecture.
538
00:36:11,370 --> 00:36:16,140
And so what we'll want to look
at is the Fourier transform as
539
00:36:16,140 --> 00:36:18,990
the pole and zero
vectors change.
540
00:36:18,990 --> 00:36:22,150
Once again, the Fourier
transform will be displayed on
541
00:36:22,150 --> 00:36:25,950
a scale from 0 to 2 pi, a
frequency scale from 0 to 2
542
00:36:25,950 --> 00:36:29,060
pi, rather than minus
pi to pi.
543
00:36:29,060 --> 00:36:32,600
And we want to observe the pole
and zero vectors as we
544
00:36:32,600 --> 00:36:34,010
sweep around the unit circle.
545
00:36:38,670 --> 00:36:42,420
We display first the pole
and zero vectors at
546
00:36:42,420 --> 00:36:44,960
omega equal to 0.
547
00:36:44,960 --> 00:36:52,140
And as the frequency increases,
the pole vector
548
00:36:52,140 --> 00:36:53,910
increases in length.
549
00:36:53,910 --> 00:36:57,640
The zero vector, since the zero
is at the origin, will
550
00:36:57,640 --> 00:37:00,190
have constant length
no matter where we
551
00:37:00,190 --> 00:37:01,430
are on the unit circle.
552
00:37:01,430 --> 00:37:04,470
Although it would affect the
phase, which we are not
553
00:37:04,470 --> 00:37:06,000
displaying here.
554
00:37:06,000 --> 00:37:09,520
And so the principle effect, the
only effect really on the
555
00:37:09,520 --> 00:37:12,540
magnitude, is due to
the pole vector.
556
00:37:12,540 --> 00:37:17,470
As the frequency continues to
increase, the pole vector
557
00:37:17,470 --> 00:37:21,890
increases in length,
monotonically in fact.
558
00:37:21,890 --> 00:37:25,190
And so that means that the
magnitude of the Fourier
559
00:37:25,190 --> 00:37:30,700
transform will decrease
monotonically until we get
560
00:37:30,700 --> 00:37:33,030
past omega equal to pi.
561
00:37:44,820 --> 00:37:48,910
Here we are now at omega
equal to pi.
562
00:37:48,910 --> 00:37:54,360
And when we continue sweeping
past this frequency around to
563
00:37:54,360 --> 00:37:59,860
2 pi, then we will see basically
the same curve swept
564
00:37:59,860 --> 00:38:01,320
out in reverse.
565
00:38:01,320 --> 00:38:04,210
Since because of the symmetry,
again, of the Fourier
566
00:38:04,210 --> 00:38:10,190
transform magnitude, sweeping
from pi to 2 pi is going to be
567
00:38:10,190 --> 00:38:13,740
equivalent with regard to the
magnitude to sweeping
568
00:38:13,740 --> 00:38:17,400
from pi back to 0.
569
00:38:17,400 --> 00:38:21,730
And so now the pole vector
begins to decrease in length
570
00:38:21,730 --> 00:38:25,140
and correspondingly, the
magnitude of the Fourier
571
00:38:25,140 --> 00:38:29,070
transform will increase.
572
00:38:29,070 --> 00:38:33,130
And that will continue until we
get around to omega equal
573
00:38:33,130 --> 00:38:36,265
to 2 pi, which is equivalent,
of course, to
574
00:38:36,265 --> 00:38:38,020
omega equal to 0.
575
00:38:38,020 --> 00:38:41,430
And obviously, if we continue
to sweep around again, we
576
00:38:41,430 --> 00:38:46,500
would simply trace out other
periods associated with the
577
00:38:46,500 --> 00:38:47,750
Fourier transform.
578
00:38:55,550 --> 00:39:00,270
Well, that hopefully gives you
kind of some feel for the
579
00:39:00,270 --> 00:39:02,760
notion of sweeping around
the unit circle.
580
00:39:02,760 --> 00:39:06,430
And of course, you can see that
because the circle is
581
00:39:06,430 --> 00:39:09,520
periodic as we go around and
around, of course, what we'll
582
00:39:09,520 --> 00:39:13,130
get is a periodic Fourier
transform, which is the way
583
00:39:13,130 --> 00:39:16,990
Fourier transforms are
supposed to be.
584
00:39:16,990 --> 00:39:26,830
Now, just as with the Laplace
transform, the region of
585
00:39:26,830 --> 00:39:30,780
convergence of the z-transform,
as we've seen in
586
00:39:30,780 --> 00:39:36,240
this example, is a very
important part of the
587
00:39:36,240 --> 00:39:39,430
specification of the
z-transform.
588
00:39:39,430 --> 00:39:42,690
And we can, in talking about
sequences and their
589
00:39:42,690 --> 00:39:46,100
transforms, either specify
the region of convergence
590
00:39:46,100 --> 00:39:49,730
implicitly, or we can specify
it explicitly.
591
00:39:49,730 --> 00:39:52,870
We can, for example, say what
it is, as let's say the
592
00:39:52,870 --> 00:39:56,020
magnitude of z being greater
than the magnitude of a.
593
00:39:56,020 --> 00:40:00,830
Or we can recognize that the
region of convergence has
594
00:40:00,830 --> 00:40:06,020
certain constraints associated
with certain properties of the
595
00:40:06,020 --> 00:40:07,480
time function.
596
00:40:07,480 --> 00:40:14,370
And in particular, there are
some important properties of
597
00:40:14,370 --> 00:40:20,060
the region of convergence which
allow us, given that we
598
00:40:20,060 --> 00:40:23,540
know certain characteristics of
the time function, to then
599
00:40:23,540 --> 00:40:26,440
identify the region of
convergence by looking at the
600
00:40:26,440 --> 00:40:27,940
pole-zero pattern.
601
00:40:27,940 --> 00:40:32,420
For example, we recognize that
the region of convergence does
602
00:40:32,420 --> 00:40:37,800
not contain any poles because of
the fact that at poles, the
603
00:40:37,800 --> 00:40:42,010
z-transform, in fact, blows up
and, of course, can't converge
604
00:40:42,010 --> 00:40:44,060
at that point.
605
00:40:44,060 --> 00:40:48,480
Furthermore, the region of
convergence consists of a ring
606
00:40:48,480 --> 00:40:52,420
in the z-plane centered
about the origin.
607
00:40:52,420 --> 00:40:55,750
Recall that with the Laplace
transform, the region of
608
00:40:55,750 --> 00:40:59,220
convergence consisted of
strips in the s-plane.
609
00:41:01,720 --> 00:41:04,380
With the z-transform, the region
of convergence consists
610
00:41:04,380 --> 00:41:07,880
of a ring, basically because of
the fact that the region of
611
00:41:07,880 --> 00:41:12,490
convergence is dependent
on the magnitude of z.
612
00:41:12,490 --> 00:41:14,990
Whereas, with the Laplace
transform, the region of
613
00:41:14,990 --> 00:41:18,310
convergence was dependent
on the real part of s.
614
00:41:18,310 --> 00:41:21,510
The fact that it's the magnitude
of z says, in
615
00:41:21,510 --> 00:41:25,580
effect, that all values of z
that have the same magnitude
616
00:41:25,580 --> 00:41:27,350
lie on a circle.
617
00:41:27,350 --> 00:41:29,900
And so the region of convergence
you would expect
618
00:41:29,900 --> 00:41:36,420
to be a concentric ring
in the z-plane.
619
00:41:36,420 --> 00:41:40,000
Furthermore, as we've already
talked about and exploited
620
00:41:40,000 --> 00:41:44,130
actually, convergence of the
Fourier transform is
621
00:41:44,130 --> 00:41:47,720
equivalent to the statement that
the region of convergence
622
00:41:47,720 --> 00:41:49,530
includes the unit circle
in the z-plane.
623
00:41:52,130 --> 00:41:57,420
Now, we can also associate the
region of convergence with
624
00:41:57,420 --> 00:42:01,700
issues about whether the
sequence is of finite duration
625
00:42:01,700 --> 00:42:04,610
or right-sided or left-sided.
626
00:42:04,610 --> 00:42:09,920
And let me sort of quickly
indicate again what the style
627
00:42:09,920 --> 00:42:12,550
of the argument is.
628
00:42:12,550 --> 00:42:17,840
If we have a finite duration
sequence, so that the sequence
629
00:42:17,840 --> 00:42:20,580
is absolutely summable, and
therefore has a Fourier
630
00:42:20,580 --> 00:42:25,590
transform that converges, then
because of the fact that it's
631
00:42:25,590 --> 00:42:30,210
0 outside some interval,
I can multiply it by an
632
00:42:30,210 --> 00:42:35,070
exponentially decaying
sequence or by an
633
00:42:35,070 --> 00:42:37,340
exponentially growing
sequence.
634
00:42:37,340 --> 00:42:41,170
And since I'm only doing this
over a finite interval, no
635
00:42:41,170 --> 00:42:45,030
matter how I choose that
exponential, we'll end up with
636
00:42:45,030 --> 00:42:47,130
an absolutely summable
product.
637
00:42:47,130 --> 00:42:51,610
So if x of n is a finite
duration, then in fact the
638
00:42:51,610 --> 00:42:56,590
region of convergence is the
entire z-plane, possibly with
639
00:42:56,590 --> 00:42:58,935
the exception of the
origin or infinity.
640
00:43:01,820 --> 00:43:06,490
On the other hand, if the
sequence is a right-sided
641
00:43:06,490 --> 00:43:12,390
sequence, then we have to be
careful that we don't multiply
642
00:43:12,390 --> 00:43:17,290
by an exponential that grows
too fast for positive time.
643
00:43:17,290 --> 00:43:22,300
Or equivalently, we might have
to choose the exponential so
644
00:43:22,300 --> 00:43:26,720
that it decays sufficiently
fast for positive time.
645
00:43:26,720 --> 00:43:31,240
As a consequence of that, for a
right-sided sequence, if we
646
00:43:31,240 --> 00:43:36,730
have a value which is in the
region of convergence, as long
647
00:43:36,730 --> 00:43:40,950
as I multiply by exponentials
that decay faster than that
648
00:43:40,950 --> 00:43:47,020
for positive time, then I'll
also have convergence.
649
00:43:47,020 --> 00:43:50,630
In other words, all finite
values of z for which the
650
00:43:50,630 --> 00:43:53,710
magnitude of z is greater than
this, so that the exponentials
651
00:43:53,710 --> 00:43:56,760
die off even faster will also
be in the region of
652
00:43:56,760 --> 00:43:59,300
convergence.
653
00:43:59,300 --> 00:44:01,810
If we combine that statement
with the fact that there are
654
00:44:01,810 --> 00:44:05,320
no poles in the region of
convergence, then we end up
655
00:44:05,320 --> 00:44:07,150
with a statement similar
to what we had
656
00:44:07,150 --> 00:44:09,000
with the Laplace transform.
657
00:44:09,000 --> 00:44:14,430
Here, the statement is that if
the sequence is right-sided,
658
00:44:14,430 --> 00:44:19,160
then the region of convergence
has to be outside the
659
00:44:19,160 --> 00:44:20,560
outermost pole.
660
00:44:20,560 --> 00:44:23,610
Essentially, because it has to
be outside someplace and can't
661
00:44:23,610 --> 00:44:24,860
include any poles.
662
00:44:27,220 --> 00:44:36,110
Finally, if we have a left-sided
sequence, then if
663
00:44:36,110 --> 00:44:39,230
we have a value which is in the
region of convergence, all
664
00:44:39,230 --> 00:44:42,390
values for which the magnitude
of z is less than that will
665
00:44:42,390 --> 00:44:44,630
also be in the region
of convergence.
666
00:44:44,630 --> 00:44:48,680
Or if x of z is rational, then
the region of convergence must
667
00:44:48,680 --> 00:44:51,130
be inside the innermost pole.
668
00:44:51,130 --> 00:44:56,610
And finally, if we have a
two-sided sequence, then
669
00:44:56,610 --> 00:45:00,130
there's kind of a balance
between the exponential factor
670
00:45:00,130 --> 00:45:01,590
that we use.
671
00:45:01,590 --> 00:45:05,980
And so in that case, then the
region of convergence will be
672
00:45:05,980 --> 00:45:09,880
a ring in the z-plane, and
essentially will extend
673
00:45:09,880 --> 00:45:12,880
outward to a pole and
inward to a pole.
674
00:45:12,880 --> 00:45:19,700
So if we had an algebraic
expression, let's say as we
675
00:45:19,700 --> 00:45:28,150
have here, then we could
associate with that a region
676
00:45:28,150 --> 00:45:30,980
of convergence outside
this pole.
677
00:45:30,980 --> 00:45:34,770
And that would correspond to
a right-sided sequence.
678
00:45:34,770 --> 00:45:40,010
Or we can associate with it a
region of convergence, which
679
00:45:40,010 --> 00:45:45,120
is inside the innermost pole.
680
00:45:45,120 --> 00:45:48,440
And that would correspond to
a left-sided sequence.
681
00:45:48,440 --> 00:45:53,890
And the third and only other
possibility is a region of
682
00:45:53,890 --> 00:45:58,330
convergence which lies between
these two poles.
683
00:45:58,330 --> 00:46:02,980
And that would then correspond
to a two-sided sequence.
684
00:46:02,980 --> 00:46:05,070
And notice incidentally because
of where I've placed
685
00:46:05,070 --> 00:46:08,700
these poles, that this is the
only one for which the region
686
00:46:08,700 --> 00:46:11,130
of convergence includes
the unit circle.
687
00:46:11,130 --> 00:46:14,670
In other words, it's the only
one for which the Fourier
688
00:46:14,670 --> 00:46:17,220
transform converges.
689
00:46:17,220 --> 00:46:19,830
OK, now we've moved through
that fairly quickly.
690
00:46:19,830 --> 00:46:24,560
And I've emphasized the fact
that it parallels very closely
691
00:46:24,560 --> 00:46:27,570
what we did with the
Laplace transform.
692
00:46:27,570 --> 00:46:32,140
What I'd like to do is just
conclude with a discussion of
693
00:46:32,140 --> 00:46:37,600
how we get the time function
back again when we have the
694
00:46:37,600 --> 00:46:42,150
z-transform including its
region of convergence.
695
00:46:42,150 --> 00:46:45,780
Well, we can, first of all,
develop a more or less formal
696
00:46:45,780 --> 00:46:46,580
expression.
697
00:46:46,580 --> 00:46:50,910
And the algebra for this is gone
through in the text, and
698
00:46:50,910 --> 00:46:53,300
you went through something
similar to this with the
699
00:46:53,300 --> 00:46:56,510
Laplace transform in the
video course manual.
700
00:46:56,510 --> 00:46:59,350
So I won't carry through
the details.
701
00:46:59,350 --> 00:47:04,130
But basically, what we can use
to develop a formal expression
702
00:47:04,130 --> 00:47:08,320
is the fact that the z-transform
is the Fourier
703
00:47:08,320 --> 00:47:11,550
transform of the sequence
exponentially weighted.
704
00:47:11,550 --> 00:47:16,550
So we can apply the inverse
transform to that, and that
705
00:47:16,550 --> 00:47:19,720
gives us not x of n, but x of
n exponentially weighted.
706
00:47:19,720 --> 00:47:24,540
And if we track that through,
then what we'll end up with is
707
00:47:24,540 --> 00:47:26,020
an expression.
708
00:47:26,020 --> 00:47:28,950
After we've taken care of a
few of the epsilons and
709
00:47:28,950 --> 00:47:33,410
deltas, we'll end up with an
expression that expresses
710
00:47:33,410 --> 00:47:41,000
formally the sequence x of n in
terms of the z-transform,
711
00:47:41,000 --> 00:47:43,830
where this, in fact, is a
contour integral in the
712
00:47:43,830 --> 00:47:45,390
complex plane.
713
00:47:45,390 --> 00:47:49,300
And so there's a formal
expression, just as there's a
714
00:47:49,300 --> 00:47:52,280
formal expression for the
Laplace transform.
715
00:47:52,280 --> 00:47:58,900
But in fact, the more typical
procedure is to use
716
00:47:58,900 --> 00:48:03,630
essentially transformed pairs
that we know together with the
717
00:48:03,630 --> 00:48:07,350
idea of using a partial
fraction expansion.
718
00:48:07,350 --> 00:48:15,460
So if we had a z-transform as
I indicate here, and if we
719
00:48:15,460 --> 00:48:21,120
expand it out in a partial
fraction expansion, then we
720
00:48:21,120 --> 00:48:26,580
can recognize, as we did in a
similar style with the Laplace
721
00:48:26,580 --> 00:48:27,860
transform--
722
00:48:27,860 --> 00:48:33,090
we can recognize that each
term, together with the
723
00:48:33,090 --> 00:48:38,450
identified region of convergence
corresponds to an
724
00:48:38,450 --> 00:48:40,240
exponential factor.
725
00:48:40,240 --> 00:48:44,420
And so this term, together with
the fact we know that the
726
00:48:44,420 --> 00:48:49,960
magnitude of z must be greater
than 2, allows us to recognize
727
00:48:49,960 --> 00:48:54,800
this as similar to
the Example 10.1.
728
00:48:54,800 --> 00:48:58,800
And in particular then, the
sequence associated with that
729
00:48:58,800 --> 00:49:00,970
is what I indicate here.
730
00:49:00,970 --> 00:49:04,230
And for the second term,
the sequence is
731
00:49:04,230 --> 00:49:06,070
what I indicate here.
732
00:49:06,070 --> 00:49:11,260
So what we're simply doing is
using the fact that we've
733
00:49:11,260 --> 00:49:15,850
worked out the example going one
direction before, and now
734
00:49:15,850 --> 00:49:19,350
we use that together with the
partial fraction expansion to
735
00:49:19,350 --> 00:49:23,240
get the individual sequences
back again, and
736
00:49:23,240 --> 00:49:24,720
then add them together.
737
00:49:24,720 --> 00:49:27,790
There's one other method which
I'll just point to, which is
738
00:49:27,790 --> 00:49:31,370
also elaborated on a little
more in the text.
739
00:49:31,370 --> 00:49:35,860
But it's kind of the idea of
developing the inverse
740
00:49:35,860 --> 00:49:42,570
z-transform by recognizing that
this z-transform formula,
741
00:49:42,570 --> 00:49:46,900
in fact, is a power series.
742
00:49:46,900 --> 00:49:54,130
So if we take x of z and expand
it in a power series,
743
00:49:54,130 --> 00:49:59,640
then we can pick off the
values of x of n by
744
00:49:59,640 --> 00:50:01,350
identifying the individual
745
00:50:01,350 --> 00:50:02,945
coefficients in this expansion.
746
00:50:06,070 --> 00:50:11,780
And so by simply doing long
division, for example, we can
747
00:50:11,780 --> 00:50:14,750
also get the inverse
transform.
748
00:50:14,750 --> 00:50:17,650
And that, by the way,
is very useful.
749
00:50:17,650 --> 00:50:23,330
Particularly if we want to get
the inverse z-transform for a
750
00:50:23,330 --> 00:50:27,700
z-transform expression,
which is not rational.
751
00:50:27,700 --> 00:50:30,330
Now we've moved through
this fairly quickly.
752
00:50:30,330 --> 00:50:33,080
On the other hand, I've stressed
that it's very
753
00:50:33,080 --> 00:50:36,220
similar to what we went through
for the Laplace
754
00:50:36,220 --> 00:50:41,320
transform, except for a very
important difference.
755
00:50:41,320 --> 00:50:44,390
The principal difference really
being that with the
756
00:50:44,390 --> 00:50:48,580
Laplace transform, it was the
j omega axis in the s-plane
757
00:50:48,580 --> 00:50:51,300
that we focused attention on
when we were thinking about
758
00:50:51,300 --> 00:50:52,970
the Fourier transform.
759
00:50:52,970 --> 00:50:56,230
Here, the unit circle
in the z-plane plays
760
00:50:56,230 --> 00:50:58,580
an important role.
761
00:50:58,580 --> 00:51:01,900
What we'll see when we continue
this in the next
762
00:51:01,900 --> 00:51:07,250
lecture is that there are
properties of the z-transform,
763
00:51:07,250 --> 00:51:10,630
just as there were properties
of the Laplace transform.
764
00:51:10,630 --> 00:51:16,910
And those properties allow us
to develop and exploit the
765
00:51:16,910 --> 00:51:21,860
z-transform in the context of
systems describable by linear
766
00:51:21,860 --> 00:51:24,250
constant coefficient difference
equations.
767
00:51:24,250 --> 00:51:28,420
So in the next lecture, we'll
focus on some properties of
768
00:51:28,420 --> 00:51:31,970
the z-transform, and then we'll
see how to use those
769
00:51:31,970 --> 00:51:37,030
properties to help us in getting
further insight and
770
00:51:37,030 --> 00:51:38,870
working with systems
describable
771
00:51:38,870 --> 00:51:40,580
by difference equations.
772
00:51:40,580 --> 00:51:41,830
Thank you.